4 BANACH ALGEBRAS WITH INVOLUTION. STAR ALGEBRAS 341 7. (a) In a star algebra, every hermitian quasi-nilpotent element (Section 15.2, Problem 5) is zero. (b) Let A be a star algebra with unit element, and let x be an element of A which is not left-invertible. Then x*x is not invertible in the star subalgebra B of A generated by 1 and x*x, and therefore there exists a sequence (yn) of elements of B such that \\ytt\\ = 1 and \imynx*x = Q (use the Gelfand-Neumark theorem). Deduce that a n-»oo noninvertible element of A is necessarily a (left or right) topological zero-divisor. 8. Let A be a star algebra with unit element, x a normal element of A, and S = SpA(x). Then there exists a unique homomorphism 9? of the star algebra ^C(S) into A, trans- forming unit element into unit element, and such that 9(ls) = x. This homomorphism is an isometry, which sends / to <p(/)*, of ^C(S) onto the star subalgebra of A gen- erated by 1, x and x*; all the elements of this subalgebra are therefore normal. Put <p(f) =/(x), and show that if/is analytic in a neighborhood of S, then the element f(x) is equal to the element so denoted in Section 15.2, Problem 11. 9. Let A be a star algebra with unit element. Let P denote the set of self-adjoint elements of A such that SpA(*) c [0, + oo [. (a) Show that if x e A is self-adjoint and \\e — x\\ <£ 1, then x e P. If x e P and ||*II ^ 1, then ||e —#|| ^ 1. (Consider the subalgebra generated by e and x.) Show that a self-adjoint element x belongs to P if and only if \\x — \\x\\e\\ <S ||jt||. (b) Deduce from (a) that P is a closed convex cone in A, such that P n (—P) = {0}. (c) Show that the relation x*xe — P implies that x — Q. (Observe first that also xx* e — P, and by writing x = u+iv, where u and v are self-adjoint, deduce that x*x e P, whence x*x = 0.) (d) Deduce from (c) that x*x e P for all x e A. (Write x*x = u — u, where «, v are hermitian and belong to P, and uv = vu =• 0 (use Problem 8). If z = xv, show that z*ze—P, and deduce that v = 0.) Deduce that e-{• x*x is invertible in A (use Problem 8). 10. (a) Let A be a commutative Banach algebra with unit element, and let JCH->;C* be any involution on A. For each character x °f A, put x*(X) = %(#*). Show that Xi-»X* *s an involutory homeomorphism of the space X(A). (b) Let X be a metrizable compact space, and <p an involutory homeomorphism of X onto X. For each/e tfc(X), put/*(*) =/(9?(x)). Show that this defines an involu- tion on ^c(X), and that every involution on ^c(X) may be so obtained. (c) Let X be the compact subspace of R2 which is the union of the segment y = 0, — 1 ^ x ^ 2, the segment x — 0, 1 <£ y ^ 2, the circle x2 + y2 = I and the open half- disk D:y>0, x2 -\~ y2 <l. Assume that the only involutory homeomorphism of X onto X is the identity (Chapter XXIV). Let A be the Banach subalgebra of ^C(X) consisting of functions which are analytic in D. Show that the spectrum of A can be canonically identified with X, and that the algebra A has no involution other than the identity. 11. Let A be a noncommutative star algebra with unit element. (a) Show that there exists a hermitian element y e P (notation of Problem 9) such that y is invertible and y2 is not in the center Z of A. (Using Problem 9, show that if the result were false, the intersection of Z with the real vector subspace H of A con- sisting of the hermitian elements would contain a neighborhood of e in H, which would imply that A was commutative.)itive measures belonging to M£(X, v), where the space X is compact,